[1863, to Majruh:] First understand what a qasam
is. How tall is its stature? What are its hands and feet like? What kind of
complexion does it have? When you can't tell this, you'll realize that qasam
is not a body or an embodied thing. It is only hypothetical. Its presence
is only in the mind. Its existence is like that of the [mythical bird] Simurgh.
That is, it can only be named, not seen. Thus the poet says that when I became
my own qasam , then it's as if in that situation my
being is the proof of my non-being. (Arshi 296)

The idiom is, 'We don't have such-and-such a thing even
to swear by' [hamaare paas falaan shai qasam khaane ko bhii
nahii;N], or 'not even the name' [naam ko bhii nahii;N]....
And it's clear that this kind of existence that would be 'to swear by', or
'strictly nominal', is a proof of oblivion and nothingness. And there's also
the idiom, 'we swear off such-and-such a thing' [hame;N falaan
shai kii qasam hai]; that is, 'we have no relationship with it'. (180)

Our existence is proof of our nonexistence because our existence
is like the case in which something 'would be only enough to swear by' [ko))ii
chiiz qasam khaane bhar kii ho]. That is, our existence is nominal [baraa-e
naam]. (325)

FWP:

Ghalib is always happy to launch himself into the zigzag progress
of a good paradox,
and in the first line he does so with a vengeance. We can only wait-- under mushairah
performance conditions-- and speculate,
and hope for enlightenment in the second line. But in proper mushairah-verse
style, the second line too remains completely opaque, until the last possible
moment, in the rhyme-word
itself, when qasam finally pulls the whole verse together,
and reminds us with a sudden burst of amusement of the idiom, and makes us
laugh and say vaah vaah .

At least for this verse, most unusually, we have some helpful interpretive comments from the poet himself.

This one reminds me also of the question of whether the beloved's
waist exists or not, in {100,3}, and the
claim that a scar is a 'token' of a liver {138,7}.
He's also played specifically with the idiomatic pleasures of qasam
: for some examples, see {89,3}.

Note for grammar fans: Strictly speaking, we should read apnii fanaa as 'its own death' (that of the 'existence'); but of course the possessive bleeds over semantically to convey the idea of 'our own death'. For more on this relatively free usage, see {15,12}.